On the Relation between Phase-Type Distributions and Positive Systems

and Applied Analysis 3 systems in control. Let us consider single-input, single output linear time-invariant systems of the form ?̇? (t) = Ax (t) + Bu (t) , y (t) = Cx (t) . (5) The linear system in (5) is said to be a positive linear system provided that, for any nonnegative input and nonnegative initial state, the state trajectory and the output are always nonnegative. Let a transfer function H(s) = C(sI − A)B be defined by the Laplace transform of the impulse response function h(t) = C exp(At)B for t ≥ 0 and, otherwise, h(t) = 0. A triple (A, B, C) is said to be a positive realization ofH(s) in a continuous-time linear positive system if and only if A is a Metzlermatrix,B ≥ 0, andC ≥ 0. A triple (A, B, C) is denoted by a minimal realization if (A, B, C) is jointly completely controllable and completely observable. An integral function of h(t) is defined by h I (t) = ∫ t


Introduction
Positive system problems have been developed in applications areas such as biological models, production systems, and economic applications.The realization problem for positive system has extensively been considered in many research papers as [1][2][3][4][5].Many research activities and applications have been devoted to the field of phase-type distributions [6][7][8].The positive representation problem finding a Markov chain associated with phase-type distribution has considerable connections with the positive realization problem in the control theory [9].
We will discuss the relationship between phase-type representation and positive realization by using the Perron-Frobenius theorem introduced in [10][11][12][13].The Perron-Frobenius theorem is an important concept for the study of positive systems.For example, the Perron-Frobenius theorem can be used to derive a transformation from positive realization to phase-type representation.We use tools and results within the broad research area of nonnegative matrix theory, which enable us to explore the characteristics and properties of a Metzler matrix and nonnegative matrix.Metzler matrices are replaced by -matrices, in particular, by the -matrices introduced in [10][11][12][13].
The connection between phase-type and positive realization has restrictively been proved in irreducible representation cases by remarking that it can be easily simplified to an irreducible case by discarding some states [9].Under the irreducible assumption, it is proven that the positive realization can be transformed into a phase-type representation [9].We will show that a positive realization normalized by a positive number can be transformed into a phase-type representation.We modify the correspondence between positive realizations and phase-type representations under more general assumptions.We use excitable systems as a subclass of the positive systems introduced in [2,14].However, the phase-type representation has a benefit that the number of free parameters in the representation can be reduced, compared with the general positive realization.
We will discuss the properties and characteristics, such as irreducibility, excitability, transparency, and stabilization introduced in [3,8,[15][16][17][18].Excitability and transparency are similar to the reachability and observability of positive linear systems [17,18].There exist unreachable and unobservable positive states that are excitable and transparent [2,14].The properties of excitability and transparency are discussed furthermore.We will demonstrate how to discard some unnecessary states when a representation is not irreducible.

Preliminaries
Before proceeding, we introduce some basic notations.An ×  nonnegative matrix  is denoted by  ≥ 0 if its entries are nonnegative and at least one entry is positive. = [  ] is defined by a strict positive matrix (i.e.,  > 0) if all entries   > 0. The associated directed graph, (), consists of  vertices {V 1 , . . ., V  }, where   ̸ = 0 denotes an edge from V  to V  .A nonnegative matrix  is said to be reducible if there exists a permutation matrix  such that where  11 and  22 are square matrices and   is defined by the transpose of .Otherwise,  is called irreducible.It is called a Frobenius normal form of  if there exists a suitable permutation  such that   is in block triangular of the form where   is square and either irreducible or a 1 × 1 null matrix.The spectral radius of , denoted by (), is defined by the largest absolute eigenvalue of .The spectrum of , denoted by (), is defined by the set of eigenvalues of .
The dominant eigenvalue of  is called by the maximal real among the real eigenvalues of , denoted by  max ().
A matrix  ∈ R × is said to be a Metzler matrix if all of its off-diagonal elements are nonnegative.If  is a Metzler matrix, then there exist a nonnegative matrix  ≥ 0 and some  > 0 such that  =  − .The real dominant eigenvalue of  is defined by  max () = () −  if  max () ≥ Re() for all  ∈ ().There is a long stream of research dealing with -matrices and -matrices instead of Metzler matrices [10].Metzler matrices are replaced by -matrices; that is, − is a -matrix if  is a Metzler matrix.In particular, a matrix  is called an -matrix if any matrix  is expressed in the form  =  −  where  ≥ 0 and  ≥ ().
Basic definitions and results of cone theory may be needed within this paper.A set K is said to be a cone if K ⊂ K for all  ≥ 0. A cone is convex if for any two points in K it contains the line segment between them.A convex cone K is solid if the interior of K is nonempty.It is pointed if K ∩ (−K) = {0}.A closed pointed solid convex cone is called a proper cone.A cone K is said to be polyhedral if it can be expressed as the set of nonnegative combinations of a finite set of generating vectors.We adopt the notation K = Cone() if K coincides with the set of nonnegative combinations of the  ×  matrix .
We discuss the phase-type distribution for a random variable  ≥ 0 in the terms of a continuous-time Markov process.A continuous-time Markov process is defined on  + 1 finite state space.The row vector  gives the initial probabilities with  state probability   .A phase-type distribution is defined as the distribution of the time to absorption in a continuoustime Markov chain (CTMC) with one absorbing state [8].If the  + 1 state is an absorbing state and all other states are transient, we define a phase-type infinitesimal generator matrix of the Markov chain, denoted by (, , ), such that where 0 refers to the column vector, row vector, or matrix with all entries equal to zero in the case without ambiguity, e  is the corresponding row vector whose th entry is one and the others are zero, and 1 is the column vector with all entries being one.We can see that  +1 = 0 if 1 = 1, and 1 +  +1 = 1 otherwise.Phase-type distributions are commonly represented by a vector-matrix tuple (, ) that describes the transient part of the CTMC.The vector-matrix tuple (, ) is a phase-type (Markovian) representation of a phase-type distribution if and only if 1 ≤ 1 and  ≥ 0,  is a Metzler matrix with 1 ≤ 0 and 1 ̸ = 0, and  is nonsingular.
The probability density function (PDF), cumulative distribution function (CDF), and Laplace-Stieltjes transform (LST) of the PDF, respectively, are defined by where (⋅) is an expectation.The ME (matrix exponential) distribution is a generalization of the phase-type distribution.A distribution function is called an ME distribution if there exists the triple (, , ) such that () = 1 −  exp(), and there is no restriction on the elements of (, , ).
We note that these representations of phase-type distributions are equivalent to the state space realizations of linear systems in control.Let us consider single-input, single output linear time-invariant systems of the form ẋ () =  () +  () , () =  () . (5) The linear system in ( 5) is said to be a positive linear system provided that, for any nonnegative input and nonnegative initial state, the state trajectory and the output are always nonnegative.Let a transfer function () = ( − ) −1  be defined by the Laplace transform of the impulse response function ℎ() =  exp() for  ≥ 0 and, otherwise, ℎ() = 0.
A triple (, , ) is said to be a positive realization of () in a continuous-time linear positive system if and only if  is a Metzler matrix,  ≥ 0, and  ≥ 0. A triple (, , ) is denoted by a minimal realization if (, , ) is jointly completely controllable and completely observable.
An integral function of ℎ() is defined by ℎ  () = ∫  0 ℎ() and ℎ  (0) = 0. We can see that an augmented realization (, , ) of ℎ  () is defined by The augmented realization (, , ) presents a state space realization of the integral function, such as ℎ  () =  exp().The augmented realization (, , ) is closely related to representation (3).We note that the positive realization of the integration of a positive system is closely related to the representation of the phase-type distribution.

Phase-Type Representation and the Positive Realization
3.1.Some Connections.Using a generalized version of the Perron-Frobenius theorem of nonnegative matrix theory, we derived a transformation from positive realization into phase-type realization under a constraint.The Perron-Frobenius theorem asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components [10,19].The Perron-Frobenius results of reducible matrices are characterized by being weaker than those of irreducible matrices.The general Perron-Frobenius theorem for reducible matrices is introduced from the result in Chapter 8 in [19] as follows.
The solvability problem of the matrix equation ( − ) =  with constraint  ≥ 0 was originally solved by Carlson [11].Several generalized versions have been discussed by researchers [12,13,20].We note that the excitability is closely related to the existence of the strict positive solution of ( − ) =  (i.e.,  > 0) as a more general case.For a discrete-time positive system with realization (, , ), (, ) is defined to be excitable if there is an  such that ∑  =0    > 0, and (, ) is defined to be transparent if there is an  such that ∑  =0   > 0 [2,14].We consider some properties of excitability.
Consider a continuous-time system with a positive realization (, , ).Since  is a Metzler matrix, we can choose an  > 0 such that ( + ) ≥ 0 and ( + ) < 1.We can define the excitability and the transparency of continuoustime positive linear systems in a similar form to discrete-time ones.The pair (, ) is excitable if there is an integer  > 0 such that for some  > 0. The pair (, ) is transparent if there exists an  such that for some  > 0.
Proof.We can choose a sufficiently large  > 0 such that a positive matrix Ã =  +  satisfies ( Ã) = , and  > || for all  ∈ ().By using the Perron-Frobenius Theorem 1 for the augmented nonnegative matrix Ã with the order  + 1, there exists an eigenvector V ≥ 0 corresponding to  such that ÃV = V.If the last entry of V is zero, V +1 = 0, then it induces that a vector is an eigenvector of + corresponding to an eigenvalue .It contradicts the fact that all the eigenvalues of + are less than .Therefore, we have V +1 > 0.
We will show that a positive realization of continuoustime positive system can be transformed into a phase-type representation normalized by a positive number.Under the irreducible assumption, it was proven that the positive realization can be transformed into phase-type representation [9].We modify the proof as a generalized version of the correspondence between positive realizations and phase-type representation.We can see that a phase-type representation is a special positive realization with excitable constraint.Theorem 4. Consider the continuous-time positive system with the positive realization (, , ) such that (, ) is excitable, and  is an asymptotically stable and Metzler matrix.Then it is transformed into a phase-type infinitesimal generator matrix ( + ,  + ,  + ) such that where  + 1 = − + and  + 1 > 0.

Common Properties and Characteristics.
We discuss the properties and characteristics, such as stability, irreducibility, excitability, and transparency, in positive systems and phasetype distributions.A positive system with a positive realization (, , ) is said to be irreducible if (, ) is excitable and (, ) is transparent in the terminology introduced in [2,14].We note that the properties and characteristics of excitability and transparency are closely related to those of the reachability and observability of positive linear systems [17,18].A phase-type representation (, ) in whose graph all the state vertices are connected to the initial vertex and to the absorbing vertex is called irreducible [8].We note that the irreducible representation is closely related to the irreducibility of the phase-type renewal process in the Markovian point process introduced in [7,8].Renewal processes provide simple models of point processes, which may describe an ordered set of points.We consider a renewal process with a phase-type distribution for the interrenewal intervals.In [8], the phase-type representation (, ) for the distribution function () is called irreducible if  * is irreducible where an infinitesimal generator  * is defined by We may associate a Markov process with a phase-type renewal process.A renewal function denoted by () for a phase-type distribution () is defined by the expected number of renewals in the interval [0, ]; that is, () = [()], where () denotes the number of renewals.A renewal density is defined by () = ()/.The Laplace transform of the renewal density () is denoted by  * (), which is rewritten by In view of control theory, the equation in ( 11) is equivalent to the positive feedback control.Because the state space realization of the inverse system (1 −  * ()) −1 is given by , we can see that its renewal density is given by () =  exp( * )(−1), which is equal to the results in [7].For an irreducible representation (, ), the vectors  exp() and exp()(−1) are strictly positive [8].We note that these results are related to the excitability and transparency of the positive linear system.The irreducibility of a positive system can be defined in a similar manner.A positive system is irreducible if each state variable influences and is influenced by another variable [2].It is defined by an irreducible realization for the positive system with a positive realization (, , ) if  is irreducible, where  is defined by For the open-loop system (5), the associated closed loop system (positive feedback system) is given by where the linear state-feedback law is () = () and  is the constant feedback gain row vector.The closed loop system (13) is positive if and only if  is a Metzler matrix.The stabilization problem of positive systems has recently been discussed in [3,15,16].It is known an unstable open positive system (5) cannot be stabilized by linear state-feedback if the restriction on nonnegative control in the closed loop is imposed [15,16].
The properties and characteristics of excitability and transparency are closely related to the reachability and observability of positive linear systems [5,17,18].A reachable set R is the set of all points which the states approach from the origin by nonnegative inputs within finite time.It was where cl() is a closure set of  [17].An observable set S is the set of initial states in which the output is nonnegative for all  ≥ 0. The observable set S can be defined by S = { |  exp() ≥ 0, ∀ ≥ 0}.
Theorem 5 (see [17]).Let a transfer function () be a strictly proper rational function with degree n, whose realization is given by (, , ).Then, () has a positive realization ( * ,  + ,  + ) with a Metzler matrix  * =  + −  if and only if there exists a generator matrix  and  ≥ 0 such that a polyhedral cone P = cone() satisfies (1) ( + )P ⊂ P; where  ∈ R × ,  ≤ , R is a reachable set, and S is an observable set.
A positive system (, , ) is said to be reducible otherwise.When a representation is not irreducible, it can be simplified by discarding some states.Our next question is how to discard some unnecessary states when a representation is not irreducible.We discuss an order reduction algorithm of the asymptotical stable and unexcitable positive realization.Let a set ⟨⟩ = {1, . . ., }.For a column vector  with  entries, we define a support set where  11 is an  1 × 1 matrix.Furthermore, we have a reduced positive realization ( 11 ,  1 ,  1 ) such that Proof.Because  * is a Metzler matrix and asymptotical stable, there is an  such that  + =  * + is a positive matrix.Define   = ∑  =0 ( + )   with  = 1/.There is an integer  > 0 such that supp(  ) = supp( +1 ) for all  ≥ .A support set for   is defined by Z = supp(  ).Let  1 be the element number of Z.We can find a permutation matrix  such that supp(    ) = ⟨ 1 ⟩.Set  1 =    = [  11  12  21  22 ] where the size of  11 is  1 .If  21 is a nonzero matrix, then we have supp( 1     ) ̸ ⊆ ⟨ 1 ⟩, but this contradicts the definition of   .We can see that there is a permutation matrix  12  0  22 ] and supp( 1 ) ⊂ ⟨ 1 ⟩.We have R ⊂ cone( 1 ).Because  * is a Metzler matrix and  + > 0, we can see cone( 1 ) ⊂ S by the definition of S. By using Theorem 5, we can derive (16).
We discussed the method to remove unnecessary states in the unexcitable case.When the transposed realization is given by (  ,   ,   ), the concept of the transparency can be interpreted by that of the excitability.A removing method of the unnecessary state in the nontransparent case is similar to that in the unexcitable case.We illustrate the previous theorems by means of an example.
We obtain By using Theorem 6 and removing the unexcitable part, we obtain an excitable positive realization ( the augmented initial vector is defined by β = [ 0], and b = e +1 .Its probability generating function is defined by () = ( −1  − ) −1  [8].We can also discuss the realization between the DPH distributions and discrete-time positive systems in a similar manner.The discrete-time linear system is represented by where  ∈  × + ,  ∈   + , and   ∈   + .The matrix tuple (, , ) denotes the positive realization (21).In the next theorem, we show that the positive realization can be transformed into a DPH representation multiplied by a positive scalar (i.e.,  is not necessarily a probability vector).Theorem 8. Assume that a realization (, , ) is denoted by a positive realization satisfying (21) and (, ) is excitable (essential reachable) and stable.Then there is a nonsingular matrix  such that the realization ( Ã, b, β), which is defined by β = , Ã =  −1 , and b =  −1 , has the properties of the DPH representation such as b = ( − Ã)1 and  ≥ 0.
Proof.Because  is positive, stable, and excitable, the absolute values of its eigenvalues are less than 1 and we can use Lemma 2. Thus, we obtain the fact that the entries of  = ( − ) −1  are positive.A similarity transform matrix  is defined by a diagonal matrix,  = diag().Compute a new realization ( Ã, b, β).We obtain the facts that ( − Ã)1 = b ≥ 0, β ≥ 0, and  ≥ 0. Therefore, we can verify that the new realization satisfies the discrete phase-type distribution properties.
We can easily deploy the properties and characteristics, such as irreducibility, excitability, transparency, and order reduction, in the discrete domain in a similar manner as in the continuous case.We omit the detailed exploration for the discrete case in this paper.

Conclusions
We considered the relation between the positive realization and the phase-type representation in continuous time and discrete time, respectively.Using the Perron-Frobenius theorem, it was shown that a phase-type representation is a special case with excitable constraint of the positive realization.We discussed their common properties and characteristics, such as irreducibility, excitability, transparency, stabilization, and order reduction.The connection between the phasetype renewal process and the feedback control of positive system was discussed.A lot of open problems related to positive system still remain and should be addressed in future research.The communities of control and probability theory can work together on solving the remaining same open problems.

4. Discrete Phase-Type Distributions and Discrete-Time Positive Systems
=1   = 1 −   .It means that ( − )1 = .An  vector  is defined by the initial probability vector.The augmented matrix tuple ( Â, b, β) is denoted by the augmented phasetype representation where the one-step transition probability matrix Â of the corresponding DTMC can be partitioned by 11,  1 ,  1 ) such that 11 = [ −2 1 1 −1 ],  1 = [1 1],and 1 = [2 1]  ] is an ( × ) matrix grouping the transition probabilities among the transient states.A column vector  = [  ] is a positive dimensional column vector grouping the probabilities from any state to the absorbing state.Thus, we have ∑